MATHEMATICS 2 — SS 2017/18
- Lecture - Tuesday 14:00 - 15:20 and Wendesday 11:00 - 12.20, O105 (Bárta)
- Seminar - Tuesday 15.30 - 16.50, O105 (Bárta)
- Exercises - Wendesday 12.30 - 13.50, O105 (Vlasák)
- Prerequisities. Knowledge of 'high school mathematics' is required (see this document for details) and Mathematics I.
- Literature. Script to download is available (but still under construction).
Exams and grading
Students will be admitted to the oral exam only if the score from the first part (midterms, homeworks and written exam together) is at least 50 points. To pass the exam successfully, at least 20 points from the oral exam are required.
- midterm test: 10 points
- homeworks: 20 points (10 homeworks, 2 points each)
- final exam: 90 points (50 points written exam, 40 points oral exam)
Grading: The total score is obtained as the sum of the points from the oral part and the first part, where the score of the first part is reduced to 60 if it exceeds 60. The final grade depends on the total score as follows.
- 70-75.5 points ... "E"
- 76-81.5 points ... "D"
- 82-87.5 points ... "C"
- 88-93.5 points ... "B"
- 94-100 points ... "A"
Midterm test takes part approximately in the half of the semester. Students are asked to
within 30 minutes. Literature is allowed, electronic devices are prohibited. There are two attempts, the better result is taken into account. Here is Midterm 1 and its solution, Midterm 2 and its solution.
Final Exam takes part in the examination period at the end of the semester. Students have three attempts to pass the final exam. It consists of a written part and oral part.
- find (and draw) the domain of a function of several variables and compute partial derivatives
See exam details for more details concerning both parts, as well as general exam rules and the requirements (list of required definitions, theorems and proofs).
- Written part. Students have 90 minutes to solve problems on
Lecture notes and other materials are allowed, electronic devices are prohibited. A sample test is available here.
- finding extrema of a function of several variables,
- matrix computations (system of linear equations, determinant, inverse matrix), and
- antiderivative (or Riemann integral).
- Oral part follows typically the day after the written exam. The oral part tests understanding the definitions and theorems and ability to apply them.
Each student should prepare answers within approximately 40 minutes. During the oral part only pencil and paper are allowed. Then the student should present answers and should answer additional questions. Here is a sample question.
Here is the beamer presentation to download and the
list of definitions and theorems for printing (last update May 14, 2018).
Preliminary plan of lectures:
- lecture 1: Functions of several variables - analogies with functions of one variable, drawing graphs, computing partal derivatives
- lecture 2: the space Rn, open and closed sets in Rn, boundary, convergence of sequences in Rn,
- lecture 3: properties of open and closed sets, bounded sets
- lecture 4: continuous functions of several variables, Heine theorem, level sets of continuous functions
- lecture 5: compact sets in Rn, characterisation of compactness in Rn, extrema of continuous functions on compact sets
- lecture 6: partial derivatives, necessary condition for a local extremum, functions of class C1
- lecture 7: weak Lagrange theorem, tangent hyperplane, continuity of C1 functions
- lecture 8: chain rule, gradient, critical point, partial derivatives of higher order, finding extrema of functions
- lecture 9: implicit function theorem
- lecture 10: implicit function theorem
- lecture 11: Lagrange multipliers theorem
- lecture 12: finding extrema of functions
- lecture 13: concave functions, quasi-concave functions
- lecture 14: Matrix calculus - solving systems of linear equations, basic definitions
- lecture 15: matrix multiplication, matrix transpose, inverse matrix
- lecture 16: linear combination, transformation of a matrix
- lecture 17: determinant
- lecture 18: Gaussian elimination, Rouché-Fontené Theorem, Cramer's rule
- lecture 19: Antiderivative and Riemann integral - basic properties of antiderivatives, existence
- lecture 20: integration by parts, substitution
- lecture 22: polynomials and partial fractions
- lecture 24: Riemann integral - definition and basic properties
- lecture 25: Newton-Leibniz formula, substitution and integration by parts for the Riemann integral
- lecture +1: Applications of the Riemann integral
- lecture +2: (matrices as linear mappings)
Seminar and exercises
Previous homeworks (without solutions) HW1, HW2, HW3, HW4 (sketch of solution), HW5, HW6, HW7, HW8, HW9, HW10.
Problems to practice:
J. Erdman: Exercises and Problems in Calculus, pdf with exercises and results,
- Chapter 24 (partial derivatives and chain rule),
- Chapter 30 (extrema),
- Chapter 26 (determinants),
- Chapter 15 (integrals)
J. Erdman: Exercises and Problems in Linear Algebra, pdf with exercises and results,
- Chapters 1-3 (systems of linear equations, matrices determinants),
R.A.Beezer: Exercise and Solution Manual for A First Course in Linear Algebra, pdf with exercises and complete solutions,
- Chapters 1,2,3,5 (systems of equations, linear independence, inverse of a matrix, determinants)
D.A.Kouba: The Calculus Page Problems List, online exercises with complete solutions
- Beginning integral calculus (integrals),
- Problems on implicit differentiation (implicit functions),
O. Tekman: Sample Calculus Problems, pdf with exercises and complete solutions
- Part I, Ex 66-70 (integrals)
- Part II (many interesting problems on vectors, partial derivatives, extrema, implicit functions)
- 1. week: partial derivatives
- 2. week: open and closed sets, cuts and contours
- 3. week: domains of functions of several variables, partial derivatives
- 4. week: extrema of functions
- 5. week: Lagrange multipliers
- 6. week: extrema of functions
- 7. week: implicit functions
- 8. week: systems of linear equations
- 9. week: inverse matrix, determinants
- 10. week: antiderivatives - substitutions, integration by parts
- 11. week: integration of rational functions (no seminar 1.5.)
- 12. week: Newton-Leibnitz formula (no seminar 8.5.)
- 13. week: Newton-Leibnitz formula (no exersises)
- suplementary week: Newton-Leibnitz formula